This section discusses how to handle type two radicals.
You can watch a video on this topic here:
In some senses, type two radicals are the most straight forward type which is why we start with the type two instead of the type one. In short, a type two radical is ‘easy’ purely because there isn’t anything we can do when it is in this form in terms of ‘simplifying’ the radical (without manipulating the radicand). The key thing here is that you absolutely cannot simplify a type two radical! This may seem straight forward, but it is often easy to overlook this and simplify things because it seems like it should work. For example;
Doing this ‘simplification’ gets us that which we can solve to find . An astute student might want to include the option because of the , so let’s consider that possibility as well; ie which gives . Now let’s plug these “solutions” of back into the original equation to see what happens;
As we can see; in both these cases our ‘solutions’ didn’t work, this is because you cannot simplify type two radicalsmath loves wasting our time.because there is a better way to simplify these radicals. . Thus we will need a different way of solving these kinds of equations, which we will discuss later in this topic.
So, when you are confronted with the need to simplify a type two radical, your only practical approach (at least at this point) is to try and manipulate the radicand to make the type two radical into a type one radical. In short, you need to factor the radicand. If possible this is almost always the best way to go, as other techniques to ‘solve’ equations with type two radicals come with a variety of possible drawbacks.
In the next section we will be discussing how to simplify type one radicals. So once you have factored the type two radical into a type one radical, you can use the techniques in the next section to then simplify the radical.